Two straight lines

If two straight lines intersect, the vertically opposite angles are equal. So $\angle ATD=\angle BTC$ , we have

$2x+2y=4x-2y$ $\implies$ $4y=2x$ $\implies$ $x=2y$ $\color{#D61F06}(1)$

If two straight lines intersect, the sum of the two adjacent angles is two right angles $(180^\circ)$ . So, $\angle ATD+\angle DTB=180$ . We have

$2x+2y+2x-y=180$ $\implies$ $4x+y=180$ $\color{#D61F06}(2)$

Substitute $\color{#D61F06}(1)$ in $\color{#D61F06}(2)$ , we have

$4(2y)+y=180$ $\implies$ $8y+y=180$ $\implies$ $9y=180$ $\implies$ $y=20$

It follows that, $x=2(20)=40$ . Since $\angle ATD$ and $\angle BTD$ are vertically opposite angles, they are equal. So

$\angle ATD=\angle BTD=2x-y=2(40)-20=\boxed{60^\circ}$